A001844 -id:A001844 - cp's OEIS frontend

A363267 Squares (A000290) alternating with centered squares (A001844).

1, 1, 4, 5, 9, 13, 16, 25, 25, 41, 36, 61, 49, 85, 64, 113, 81, 145, 100, 181, 121, 221, 144, 265, 169, 313, 196, 365, 225, 421, 256, 481, 289, 545, 324, 613, 361, 685, 400, 761, 441, 841, 484, 925, 529, 1013, 576, 1105, 625, 1201, 676, 1301, 729, 1405, 784

Offset: 1

Clark Kimberling, May 24 2023

nonn

This is a linear recurrence sequence. If the terms are arranged in nondecreasing order, the result, A363319, is linearly recurrent. If the terms are arranged in increasing order, so that there are no duplicates, the result, A363282, is not linearly recurrent.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A001844, A123596, A363268, A363269, A363282, A363319.

c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, 2 c[n - 1] - n + 1]
Table[c[n], {n, 1, 120}]

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x*(-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x^2)^3.
a(n+1) = n/2+3*n^2/8+3/4+(-1)^n*(1/4+n/2-n^2/8). - R. J. Mathar, Jun 15 2023

A190406 Decimal expansion of Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).

Original entry on oeis.org

5, 3, 1, 3, 7, 2, 1, 0, 0, 1, 1, 5, 2, 7, 7, 1, 3, 5, 4, 7, 9, 8, 7, 9, 8, 5, 8, 9, 6, 2, 5, 5, 3, 5, 3, 1, 7, 1, 2, 8, 4, 3, 2, 0, 1, 8, 6, 2, 0, 6, 6, 3, 9, 4, 0, 7, 8, 8, 8, 7, 1, 6, 1, 3, 5, 7, 8, 9, 0, 6, 8, 8, 0, 2, 3, 7, 7, 6, 0, 4, 7, 6, 0, 7, 3, 0, 3, 4, 5, 7, 7, 9, 6, 0, 7, 1, 2, 3, 4, 9, 2, 0, 6, 1, 0, 7, 1, 1, 5, 2, 2, 0, 6, 3, 9, 0, 0, 7, 3, 5

Offset: 0

Clark Kimberling, May 10 2011

See A190404.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A190404-A190407.

evalf(JacobiTheta2(0,1/4)/2^(3/2)) ; # R. J. Mathar, Jul 15 2013

(See A190404.)
(* or *) RealDigits[EllipticTheta[2, 0, 1/4]/(2*Sqrt[2]), 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)

th2(x)=x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(1/4)/sqrt(8) \\ Charles R Greathouse IV, Jun 06 2016

def A190406(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*j*(j+1)+1) for j in range(0,b)]),b)
A190406(409) # Danny Rorabaugh, Mar 26 2015

a(n) = floor(10^(n+1)*Sum_{j>=0} (1/2)^(2*j*(j+1)+1)) mod 10. - Danny Rorabaugh, Mar 26 2015

A253460 Indices of centered heptagonal numbers (A069099) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 16, 112, 3937, 28321, 999856, 7193296, 253959361, 1827068737, 64504677712, 464068265776, 16383934179361, 117871512438241, 4161454776879856, 29938900091047312, 1056993129393303937, 7604362751613578881, 268472093411122320016, 1931478200009757988336

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers y in the solutions to 4*x^2 - 7*y^2 - 4*x + 7*y = 0, the corresponding values of x being A253459.

			16 is in the sequence because the 16th centered heptagonal number is 841, which is also the 21st centered square number.

Colin Barker, Table of n, a(n) for n = 1..832
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Cf. A001844, A069099, A253459, A253599.

Vec(-x*(x^4+15*x^3-158*x^2+15*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+15*x^3-158*x^2+15*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).
a(n) = A105040(n) + 1. - Michel Marcus, Mar 12 2024

A253475 Indices of centered square numbers (A001844) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 6, 55, 540, 5341, 52866, 523315, 5180280, 51279481, 507614526, 5024865775, 49741043220, 492385566421, 4874114620986, 48248760643435, 477613491813360, 4727886157490161, 46801248083088246, 463284594673392295, 4586044698650834700, 45397162391834954701

Offset: 1

Colin Barker, Jan 02 2015

Also positive integers x in the solutions to 4*x^2 - 6*y^2 - 4*x + 6*y = 0, the corresponding values of y being A054318.
Also indices of centered hexagonal numbers (A003215) which are also hexagonal numbers (A000384).
Also indices of terms in sequence A193218 which are the square root of a sum of 5th powers (A000539). - Daniel Poveda Parrilla, Jun 10 2017

			6 is in the sequence because the 6th centered square number is 61, which is also the 5th centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A000384, A000539, A001844, A003215, A054318, A193218, A253175.

LinearRecurrence[{11, -11, 1}, {1, 6, 55}, 25] (* Paolo Xausa, May 30 2025 *)

Vec(x*(5*x-1)/((x-1)*(x^2-10*x+1)) + O(x^100))

a(n) = 11*a(n-1)-11*a(n-2)+a(n-3).
G.f.: x*(5*x-1) / ((x-1)*(x^2-10*x+1)).
a(n) = sqrt((-2-(5-2*sqrt(6))^n-(5+2*sqrt(6))^n)*(2-(5-2*sqrt(6))^(1+n)-(5+2*sqrt(6))^(1+n)))/(4*sqrt(2)). - Gerry Martens, Jun 04 2015
2*a(n) = 1+A054320(n-1). - R. J. Mathar, Feb 07 2022

A308215 a(n) is the multiplicative inverse of A001844(n+1) modulo A001844(n); where A001844 is the sequence of centered square numbers.

Original entry on oeis.org

0, 2, 12, 11, 39, 28, 82, 53, 141, 86, 216, 127, 307, 176, 414, 233, 537, 298, 676, 371, 831, 452, 1002, 541, 1189, 638, 1392, 743, 1611, 856, 1846, 977, 2097, 1106, 2364, 1243, 2647, 1388, 2946, 1541, 3261, 1702, 3592, 1871, 3939, 2048

Offset: 0

Daniel Hoyt, May 15 2019

nonn

The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A054552 (a number spiral arm) and (A001844-n). The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.

Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844

Cf. A001844, A033951, A054552, A308217.

f(n) = 2*n*(n+1)+1; \\ A001844
a(n) = lift(1/Mod(f(n+1), f(n))); \\ Michel Marcus, May 16 2019

import gmpy2
sos = [] # sum of squares
a=0
b=1
for i in range(50):
    c = a**2 + b**2
    sos.append(c)
    a +=1
    b +=1
ls = []
for i in range(len(sos)-1):
    c = gmpy2.invert(sos[i+1],sos[i])
    ls.append(int(c))
print(ls)

a(n) satisfies a(n)*(2*n*(n+1)+1) == 1 (mod 2*n*(n-1)+1).
Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x*(2 + 12*x + 5*x^2 + 3*x^3 + x^4 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = (3 + (-1)^n + 2*(2+(-1)^n)*n + 2*(3+(-1)^n)*n^2) / 4 for n>0.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
(End)

A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.

Original entry on oeis.org

1, 8, 2, 23, 3, 46, 4, 77, 5, 116, 6, 163, 7, 218, 8, 281, 9, 352, 10, 431, 11, 518, 12, 613, 13, 716, 14, 827, 15, 946, 16, 1073, 17, 1208, 18, 1351, 19, 1502, 20, 1661, 21, 1828, 22, 2003, 23, 2186, 24, 2377, 25, 2576, 26, 2783, 27, 2998, 28, 3221, 29, 3452

Offset: 0

Daniel Hoyt, May 15 2019

The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A033951 (a number spiral arm) and the natural numbers. The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.

Robert Israel, Table of n, a(n) for n = 0..10000
Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A001844, A054552, A033951, A308215.

A001844:= n -> 2*n*(n+1)+1:
seq(1/A001844(n) mod A001844(n+1),n=0..100); # Robert Israel, Aug 11 2019

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 8, 2, 23, 3, 46}, 30] (* Georg Fischer, Dec 06 2024 *)

f(n) = 2*n*(n+1)+1; \\ A001844
a(n) = lift(1/Mod(f(n), f(n+1))); \\ Michel Marcus, May 16 2019

a(n) satisfies a(n)*(2*n*(n-1)+1) == 1 (mod 2*n*(n+1)+1).
Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: (1 + 8*x - x^2 - x^3 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
a(n) = (9 - 5*(-1)^n + (8-6*(-1)^n)*n - 2*(-1+(-1)^n)*n^2) / 4. (End)
From Robert Israel, Aug 11 2019: (Start)
a(n) = 1 + n/2 if n is even, since 0 < 1+n/2 < A001844(n+1) and (1+n/2)*A001844(n)-1 = (n/2)*A001844(n+1).
a(n) = n^2 + 7/2*(n+1) if n is odd, since 0 < n^2+7/2*(n+1) < A001844(n+1) and (n^2+7/2*(n+1))*A001844(n)-1 = (n^2+3*k/2+1/2)*A001844(n+1).
Colin Barker's conjectures easily follow. (End)
E.g.f.: ((2 + 9*x)*cosh(x) + (7 + x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Dec 06 2024

A363282 Squares (A000290) and centered squares (A001844), in increasing order (i.e., sorted and without duplicates).

Original entry on oeis.org

1, 4, 5, 9, 13, 16, 25, 36, 41, 49, 61, 64, 81, 85, 100, 113, 121, 144, 145, 169, 181, 196, 221, 225, 256, 265, 289, 313, 324, 361, 365, 400, 421, 441, 481, 484, 529, 545, 576, 613, 625, 676, 685, 729, 761, 784, 841, 900, 925, 961, 1013, 1024, 1089, 1105

Offset: 1

Clark Kimberling, May 25 2023

nonn

This sequence consists of the numbers in A363267 arranged in increasing order. Unlike A363267, this is not a linear recurrence sequence; see A363319.

Cf. A000290, A001844, A363267, A363319.

c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, 2 c[n - 1] - n + 1]
u = Table[c[n], {n, 1, 120}]  (* A363267 *)
Union[u] (* A363282 *)

Definition corrected by N. J. A. Sloane, Jun 12 2023

A253459 Indices of centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 21, 148, 5208, 37465, 1322685, 9515836, 335956656, 2416984753, 85331667813, 613904611300, 21673907667720, 155929354285321, 5505087215932941, 39605442083860108, 1398270478939299168, 10059626359946181985, 355155196563366055605, 2555105489984246363956

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers x in the solutions to 4*x^2 - 7*y^2 - 4*x + 7*y = 0, the corresponding values of y being A253460.

			21 is in the sequence because the 21st centered square number is 841, which is also the 16th centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..832
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Cf. A001844, A069099, A253460, A253599.

Vec(x*(20*x^3+127*x^2-20*x-1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(20*x^3+127*x^2-20*x-1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

A253599 Centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 841, 43513, 54236113, 2807177521, 3498988573081, 181102250526121, 225733748749491361, 11683630587634972513, 14562987063325697070313, 753757743549580366157401, 939516547177660272044661361, 48627927055673997154643575441, 60611970510056587727363585953081

Offset: 1

Colin Barker, Jan 05 2015

			841 is in the sequence because it is the 21st centered square number and the 16th centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..416
Index entries for linear recurrences with constant coefficients, signature (1,64514,-64514,-1,1).

Cf. A001844, A069099, A253459, A253460.

LinearRecurrence[{1,64514,-64514,-1,1},{1,841,43513,54236113,2807177521},20] (* Harvey P. Dale, Mar 26 2023 *)

Vec(-x*(x^4+840*x^3-21842*x^2+840*x+1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+840*x^3-21842*x^2+840*x+1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

A254228 Indices of heptagonal numbers (A000566) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 41, 185, 13105, 59473, 4219673, 19150025, 1358721505, 6166248481, 437504104841, 1985512860761, 140874963037201, 639328974916465, 45361300593873785, 205861944410240873, 14606197916264321473, 66286906771122644545, 4703150367736517640425

Offset: 1

Colin Barker, Jan 27 2015

Also positive integers x in the solutions to 5*x^2 - 4*y^2 - 3*x + 4*y - 2 = 0, the corresponding values of y being A254229.

			41 is in the sequence because the 41st heptagonal number is 4141, which is also the 46th centered square number.

Colin Barker, Table of n, a(n) for n = 1..797
Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Cf. A000566, A001844, A254229, A254230.

Vec(-x*(x^4+40*x^3-178*x^2+40*x+1)/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)) + O(x^100))

a(n) = a(n-1)+322*a(n-2)-322*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+40*x^3-178*x^2+40*x+1) / ((x-1)*(x^2-18*x+1)*(x^2+18*x+1)).

cp's OEIS Frontend

A363267 Squares (A000290) alternating with centered squares (A001844).

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A190406 Decimal expansion of Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).

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A253460 Indices of centered heptagonal numbers (A069099) which are also centered square numbers (A001844).

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A253475 Indices of centered square numbers (A001844) which are also centered hexagonal numbers (A003215).

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A308215 a(n) is the multiplicative inverse of A001844(n+1) modulo A001844(n); where A001844 is the sequence of centered square numbers.

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A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.

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A363282 Squares (A000290) and centered squares (A001844), in increasing order (i.e., sorted and without duplicates).

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A253459 Indices of centered square numbers (A001844) which are also centered heptagonal numbers (A069099).