A085787 -id:A085787 - cp's OEIS frontend

A080860 a(n) = 10n^2 + 5n + 1.

1, 16, 51, 106, 181, 276, 391, 526, 681, 856, 1051, 1266, 1501, 1756, 2031, 2326, 2641, 2976, 3331, 3706, 4101, 4516, 4951, 5406, 5881, 6376, 6891, 7426, 7981, 8556, 9151, 9766, 10401, 11056, 11731, 12426, 13141, 13876, 14631, 15406, 16201, 17016, 17851

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Column T(n,5) of A080853.
Sequence found by reading the line from 1, in the direction 1, 16, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

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

Cf. A080853, A085787.

Table[10n^2+5n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,16,51},50] (* Harvey P. Dale, Aug 05 2014 *)

a(n)=10*n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(5,0) + C(5,1)*n + C(5,2)*n^2.
G.f.: (C(4,0) + (C(6,2) - 2)*x + C(4,2)*x^2)/(1-x)^3 = (1 + 13*x + 6*x^2)/(1-x)^3.
a(n) = 20*n + a(n-1) - 5 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(1 + 15*x + 10*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
Sum_{n>=0} a(n)/n! = 26*e. - Davide Rotondo, Feb 15 2025

Definition replaced with the closed form by Bruno Berselli, Jan 16 2013

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

A292850 Lucas numbers that are also generalized heptagonal numbers.

Original entry on oeis.org

1, 4, 7, 18

Offset: 1

Tomohiro Yamada, Sep 25 2017

Intersection of A000032 and A085787.
Except 4, these are also ordinary heptagonal numbers.
All terms are shown, as confirmed by Srinivasa Rao (2002).
All (generalized) g-gonal numbers in Lucas sequences up to g=20 have been determined, see Tengely (2009).

B. Srinivasa Rao, Heptagonal Numbers in the Lucas Sequence and Diophantine Equations x^2(5x-3)^2 = 20y^2+-16, The Fibonacci Quarterly, Vol. 40, No. 4 (2002), 319-322.
Szabolcs Tengely, Finding g-gonal numbers in recurrence sequences, Fibonacci Quarterly, vol.46/47, no.3, pp.235-240, (2009).

Cf. A000032, A085787.

Cf. A248506 (triangular Lucas Numbers).

A377224 Number of ways to write n as x(5x+1) + y(5y+1)/2 + z(5z+1)/2, where x,y,z are integers with y(5y+1) <= z(5z+1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 1, 2, 3, 2, 3, 2, 2, 1, 3, 1, 3, 4, 1, 3, 2, 4, 2, 6, 2, 4, 5, 4, 3, 5, 3, 3, 4, 2, 2, 4, 1, 3, 3, 3, 3, 7, 1, 6, 6, 6, 3, 8, 4, 3, 7, 3, 7, 4, 4, 2, 4, 1, 5, 6, 1, 6, 7, 4, 4, 9, 6, 5, 8, 3, 6, 5, 3, 4, 5, 3, 3, 4, 1, 9, 6, 5, 3, 9, 5, 6, 9, 6, 8, 10, 3, 3, 9, 4, 7, 7, 4, 7, 5, 4

Offset: 0

Zhi-Wei Sun, Nov 13 2024

nonn

Conjecture 1: a(n) = 0 only for n = 1. Also, a(n) = 1 only for n = 0, 2, 3, 5, 7, 14, 16, 19, 37, 43, 58, 61, 79.
This has been verified for n <= 2*10^6.
Conjecture 2: Let N be the set of all nonnegative integers. Then
{x*(5*x+1) + y*(5*y+1)/2 + 5*z*(5*z+1)/2: x,y,z are integers} = N\{1,5},
{x*(5*x+1) + y*(5*y+1)/2 + 3*z*(5*z+1)/2: x,y,z are integers} = N\{1,5,32},
{x*(5*x+1) + y*(5*y+1)/2 + 2*z*(5*z+1): x,y,z are integers} = N\{1,5,70},
and
{x*(5*x+1)/2 + y*(5*y+1)/2 + z*(5*z+1)/2: x,y,z are integers} = N\{1,10,19,94}.
Conjecture 3: We have
{x*(5*x+3) + y*(5*y+3)/2 + 3*z*(5*z+3)/2: x,y,z are integers} = N\{31,77},
{x*(5*x+3) + y*(5*y+3)/2 + 5*z*(5*z+3): x,y,z are integers} = N\{10,16},
and
{x*(5*x+3)/2 + y*(5*y+3)/2 + 5*z*(5*z+3)/2: x,y,z are integers} = N\{3,15,29,44}.

			a(14) = 1 with 14 = 0*(5*0+1) + 1*(5*1+1)/2 + 2*(5*2+1)/2.
a(37) = 1 with 37 = (-1)*(5*(-1)+1) + (-2)*(5*(-2)+1)/2 + 3*(5*3+1)/2.
a(58) = 1 with 58 = (-2)*(5*(-2)+1) + (-1)*(5*(-1)+1)/2 + (-4)*(5*(-4)+1)/2.
a(79) = 1 with 79 = -4*(5*(-4)+1) + 0*(5*0+1)/2 + 1*(5*1+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162 (2016), 190-211.
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.
Zhi-Wei Sun, New results similar to Lagrange's four-square theorem, arXiv:2411.14308 [math.NT], 2024.

Cf. A057569, A085787, A306383.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[40(n-x(5x+1)-y(5y+1)/2)+1],r=r+1],{x,-Floor[(Sqrt[20n+1]+1)/10],(Sqrt[20n+1]-1)/10},{y,-Floor[(Sqrt[20(n-x(5x+1))+1]+1)/10],Floor[(Sqrt[20(n-x(5x+1))+1]-1)/10]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A235670 Square array read by antidiagonals upwards in which the n-th column gives the partial sums of the n-th column of A211970.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 14, 7, 3, 2, 1, 24, 12, 5, 3, 2, 1, 40, 19, 8, 4, 3, 2, 1, 64, 30, 12, 6, 4, 3, 2, 1, 100, 45, 17, 9, 5, 4, 3, 2, 1, 154, 67, 24, 13, 7, 5, 4, 3, 2, 1, 232, 97, 34, 17, 10, 8, 6, 5, 4, 3, 2, 1, 344, 139, 47, 22, 14, 8, 6, 5, 4, 3, 2, 1

Offset: 0

Omar E. Pol, Jan 13 2014

The column 0 is related to A008794 in the same way as the column k is related to the generalized (k+4)-gonal numbers, for k >= 1. For more information see A195152 and A211970.

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1,...
2,     2,   2,   2,   2,   2,  2,  2,  2,  2,  2,...
4,     4,   3,   3,   3,   3,  3,  3,  3,  3,  3,...
8,     7,   5,   4,   4,   4,  4,  4,  4,  4,  4,...
14,   12,   8,   6,   5,   5,  5,  5,  5,  5,  5,...
24,   19,  12,   9,   7,   6,  6,  6,  6,  6,  6,...
40,   30,  17,  13,  10,   8,  7,  7,  7,  7,  7,...
64,   45,  24,  17,  14,  11,  9,  8,  8,  8,  8,...
100,  67,  34,  22,  18,  15, 12, 10,  9,  9,  9,...
154,  97,  47,  29,  22,  19, 16, 13, 11, 10, 10,...
232, 139,  63,  39,  27,  23, 20, 17, 14, 12, 11,...
344, 195,  84,  51,  34,  27, 24, 21, 18, 15, 13,...
504, 272, 112,  65,  44,  32, 28, 25, 22, 19, 16,...
728, 383, 147,  81,  56,  39, 32, 29, 26, 23, 20,...
...

Column 1 is A015128, the partial sums of A211971.
Column 2 is A000070, the partial sums of A000041.
Column 3 is A233969, the partial sums of A006950.
Cf. A000217, A001318, A008794, A085787, A057077, A195152, A211970.

T(n,k) = Sum_{j=0..n} A211970(j,k), (n>=0, k>=0).

A292472 Generalized heptagonal numbers that are also Fibonacci numbers.

Original entry on oeis.org

0, 1, 13, 34, 55

Offset: 1

Felix Fröhlich, Sep 17 2017

Intersection of A000045 and A085787.
Exactly five such numbers exist (cf. Srinivasa Rao, 2003).
All (generalized) g-gonal numbers in Fibonacci sequences up to g=20 have been determined (cf. Tengely, 2009). - Tomohiro Yamada, Sep 26 2017

B. Srinivasa Rao, Heptagonal Numbers in Fibonacci Sequence and Diophantine Equations 4x^2 = 5y^2(5y-3)^2+-16, The Fibonacci Quarterly, Vol. 41, No. 5 (2003), 414-420.
Szabolcs Tengely, Finding g-gonal numbers in recurrence sequences, Fibonacci Quarterly, Vol.46/47, No. 3 (2009), 235-240.

Cf. A000045, A085787.

Cf. A292850 (Generalized heptagonal Lucas numbers).

Intersection[Array[(# (# + 1)/2 - 1)/5 &, 50, 0], Array[Fibonacci, 50, 0]] (* Michael De Vlieger, Sep 18 2017 *)

a085787(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2
is_a000045(n) = my(x=0); while(fibonacci(x) < n, x++); if(fibonacci(x)==n, return(1)); 0
for(n=0, 60, if(is_a000045(a085787(n)), print1(a085787(n), ", ")))

cp's OEIS Frontend

A080860 a(n) = 10*n^2 + 5*n + 1.

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A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

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A292850 Lucas numbers that are also generalized heptagonal numbers.

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A377224 Number of ways to write n as x*(5*x+1) + y*(5*y+1)/2 + z*(5*z+1)/2, where x,y,z are integers with y*(5*y+1) <= z*(5*z+1).

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A235670 Square array read by antidiagonals upwards in which the n-th column gives the partial sums of the n-th column of A211970.

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A292472 Generalized heptagonal numbers that are also Fibonacci numbers.

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PARI

A080860 a(n) = 10n^2 + 5n + 1.

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).

A377224 Number of ways to write n as x(5x+1) + y(5y+1)/2 + z(5z+1)/2, where x,y,z are integers with y(5y+1) <= z(5z+1).