A263942 -id:A263942 - cp's OEIS frontend

A071954 a(n) = 4*a(n-1) - a(n-2) - 4, with a(0) = 2, a(1) = 4.

2, 4, 10, 32, 114, 420, 1562, 5824, 21730, 81092, 302634, 1129440, 4215122, 15731044, 58709050, 219105152, 817711554, 3051741060, 11389252682, 42505269664, 158631825970, 592022034212, 2209456310874, 8245803209280, 30773756526242, 114849222895684

Offset: 0

Lekraj Beedassy, Jun 25 2002

a(n) gives the side of a cube having a square number of cubes in its two outermost layers, i.e., solutions p to the equation p^3 - (p - 4)^3 = q^2. The corresponding q is given by 4*A001075(n).

			G.f. = 2 + 4*x + 10*x^2 + 32*x^3 + 114*x^4 + 420*x^5 + 1562*x^6 + ...

M. E. Larsen, "Four Cubes" in Puzzler's Tribute, Ed. D. Wolfe & T. Rodgers, pp. 69-70, A. K. Peters, MA, 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Reinhard Zumkeller)
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Equals A052530(n) + 2, n > 0.

Cf. A001075, A003699, A072110, A263942.

a:=[2,4,10];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 25 2019

a071954 n = a071954_list !! n
a071954_list = 2 : 4 : zipWith (-)
               (map ((4 *) . pred) (tail a071954_list)) a071954_list
-- Reinhard Zumkeller, Aug 11 2011

I:=[2,4,10]; [n le 3 select I[n] else 5*Self(n-1) -5*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 25 2019

a[n_]:= a[n] = 4*a[n-1] -a[n-2] -4; a[0]=2; a[1]=4; Table[a[n], {n,0,30}]
LinearRecurrence[{5,-5,1},{2,4,10},30] (* Harvey P. Dale, May 05 2011 *)

Vec((2-6*x)/(1-5*x+5*x^2-x^3)+O(x^30)) \\ Charles R Greathouse IV, Feb 09 2012

{a(n) = my(w=quadgen(12)); simplify( 2 + ((2+w)^n - (2-w)^n) / w)}; /* Michael Somos, Nov 03 2016 */

(2*(1-3*x)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 2, with a(0) = 2, a(1) = 4, a(2) = 10.
G.f.: 2*(1 - 3*x)/((1-x)*(1 -4*x +x^2)). - Harvey P. Dale, May 05 2011
a(n) = (2 + (-(2 - sqrt(3))^n + (2 + sqrt(3))^n)/sqrt(3)). - Colin Barker, Nov 03 2016
A263942(n) = -a(-1-n) for all n in Z. - Michael Somos, Nov 03 2016
E.g.f.: (2/3)*(3*exp(x) + sqrt(3)*exp(2*x)*sinh(sqrt(3)*x)). - Franck Maminirina Ramaharo, Nov 14 2018
From G. C. Greubel, Feb 25 2019: (Start)
a(n) = 2*A072110(n).
a(n) = 2*(1 - (-i)^(n+1)*F(n, 4*i)), where i=sqrt(-1) and F(n,x) is the Fibonacci polynomial. (End)

Edited by Robert G. Wilson v, Jun 27 2002

A263943 Positive integers n such that (n+21)^3 - n^3 is a square.

Original entry on oeis.org

7, 119, 4564, 32900, 1161895, 8359127, 295119412, 2123188004, 74959171399, 539281396535, 19039334418580, 136975351534532, 4835915983150567, 34791200008377239, 1228303620385828084, 8836827826776286820, 311984283662017185415, 2244519476801168477687

Offset: 1

Colin Barker, Oct 30 2015

			7 is in the sequence because (7+21)^3 - 7^3 = 147^2.

Colin Barker, Table of n, a(n) for n = 1..831
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Cf. A263942 (4), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{1,254,-254,-1,1},{7,119,4564,32900,1161895},20] (* Harvey P. Dale, Jan 11 2017 *)

Vec(7*x*(4*x^4+16*x^3-381*x^2-16*x-1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^30))

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

G.f.: 7*x*(4*x^4+16*x^3-381*x^2-16*x-1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

A263944 Positive integers n such that (n+28)^3 - n^3 is a square.

Original entry on oeis.org

28, 189, 959, 4648, 22323, 107009, 512764, 2456853, 11771543, 56400904, 270233019, 1294764233, 6203588188, 29723176749, 142412295599, 682338301288, 3269279210883, 15664057753169, 75051009555004, 359590990021893, 1722903940554503, 8254928712750664

Offset: 1

Colin Barker, Oct 30 2015

			189 is in the sequence because (189+28)^3 - 189^3 = 1862^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6,1).

Cf. A263942 (4), A263943 (21), A263945 (39), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{6,-6,1},{28,189,959},30] (* Harvey P. Dale, Dec 14 2016 *)

Vec(7*x*(x-4)*(x+1)/((x-1)*(x^2-5*x+1)) + O(x^40))

a(n) = 6*a(n-1)-6*a(n-2)+a(n-3) for n>3.
G.f.: 7*x*(x-4)*(x+1) / ((x-1)*(x^2-5*x+1)).
a(n) = 7*(-2+(2^(-1-n)*((5-sqrt(21))^n*(-7+sqrt(21))+(5+sqrt(21))^n*(7+sqrt(21))))/sqrt(21)). - Colin Barker, Mar 05 2016

A263945 Positive integers n such that (n+39)^3 - n^3 is a square.

Original entry on oeis.org

26, 871, 59930, 1155895, 77814386, 1500376111, 101003038370, 1947487061455, 131101866015146, 2527836705417751, 170170121084646410, 3281130096145204615, 220880686066005050306, 4258904336959770197791, 286702960343553470676050, 5528054548243685571553375

Offset: 1

Colin Barker, Oct 30 2015

			26 is in the sequence because (26+39)^3 - 26^3 = 507^2.

Colin Barker, Table of n, a(n) for n = 1..642
Index entries for linear recurrences with constant coefficients, signature (1,1298,-1298,-1,1).

Cf. A263942 (4), A263943 (21), A263944 (28), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{1,1298,-1298,-1,1},{26,871,59930,1155895,77814386},20] (* Harvey P. Dale, Mar 25 2020 *)

Vec(13*x*(5*x^4+65*x^3-1947*x^2-65*x-2)/((x-1)*(x^2-36*x-1)*(x^2+36*x-1)) + O(x^30))

a(n) = a(n-1)+1298*a(n-2)-1298*a(n-3)-a(n-4)+a(n-5) for n>5.

G.f.: 13*x*(5*x^4+65*x^3-1947*x^2-65*x-2) / ((x-1)*(x^2-36*x-1)*(x^2+36*x-1)).

A263946 Positive integers n such that (n+52)^3 - n^3 is a square.

Original entry on oeis.org

26, 2626, 132522, 6624722, 331104826, 16548617826, 827099787722, 41338440769522, 2066094938689626, 103263408493713026, 5161104329746962922, 257951953078854434322, 12892436549612974754426, 644363875527569883288226, 32205301339828881189658122

Offset: 1

Colin Barker, Oct 30 2015

			26 is in the sequence because (26+52)^3 - 26^3 = 676^2.

Colin Barker, Table of n, a(n) for n = 1..588
Index entries for linear recurrences with constant coefficients, signature (51,-51,1).

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{51,-51,1},{26,2626,132522},20] (* Harvey P. Dale, Feb 05 2019 *)

Vec(26*x*(3*x^2-50*x-1)/((x-1)*(x^2-50*x+1)) + O(x^30))

a(n) = 51*a(n-1)-51*a(n-2)+a(n-3) for n>3.
G.f.: 26*x*(3*x^2-50*x-1) / ((x-1)*(x^2-50*x+1)).
a(n) = 26*(-6-(6+sqrt(39))*(25+4*sqrt(39))^(-n)+(-6+sqrt(39))*(25+4*sqrt(39))^n)/6. - Colin Barker, Mar 03 2016

A263947 Positive integers n such that (n+57)^3 - n^3 is a square.

Original entry on oeis.org

551, 13471, 67002512, 1560515752, 7745359676111, 180392503180711, 895348087775371352, 20853012581126608912, 103500448242912021166871, 2410566548172681237123151, 11964444815088795735075876992, 278656671814812593067838694872, 1383065891631134161140389210648831

Offset: 1

Colin Barker, Oct 30 2015

			551 is in the sequence because (551+57)^3 - 551^3 = 7581^2.

Colin Barker, Table of n, a(n) for n = 1..394
Index entries for linear recurrences with constant coefficients, signature (1,115598,-115598,-1,1).

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{1, 115598, -115598, -1, 1}, {551, 13471, 67002512, 1560515752, 7745359676111}, 15] (* Paolo Xausa, Mar 05 2024 *)

Vec(19*x*(32*x^4+680*x^3-173397*x^2-680*x-29)/((x-1)*(x^2-340*x+1)*(x^2+340*x+1)) + O(x^20))

a(n) = a(n-1)+115598*a(n-2)-115598*a(n-3)-a(n-4)+a(n-5) for n>5.

G.f.: 19*x*(32*x^4+680*x^3-173397*x^2-680*x-29) / ((x-1)*(x^2-340*x+1)*(x^2+340*x+1)).

A263948 Positive integers n such that (n+61)^3 - n^3 is a square.

Original entry on oeis.org

244, 267607, 260678620, 253900737919, 247299058084132, 240869028673236295, 234606186628674096844, 228506184907299897119407, 222564789493523471120235220, 216777876460506953571212014519, 211141429107744279254889381935932, 205651535173066467487308686793612895

Offset: 1

Colin Barker, Oct 30 2015

			244 is in the sequence because (244+61)^3 - 244^3 = 3721^2.

Colin Barker, Table of n, a(n) for n = 1..334
Index entries for linear recurrences with constant coefficients, signature (975,-975,1).

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{975, -975, 1}, {244, 267607, 260678620}, 15] (* Paolo Xausa, Mar 05 2024 *)

Vec(61*x*(5*x^2-487*x-4)/((x-1)*(x^2-974*x+1)) + O(x^15))

a(n) = 975*a(n-1)-975*a(n-2)+a(n-3) for n>3.

G.f.: 61*x*(5*x^2-487*x-4) / ((x-1)*(x^2-974*x+1)).

A263949 Positive integers n such that (n+84)^3 - n^3 is a square.

Original entry on oeis.org

28, 476, 1106, 8218, 18256, 131600, 291578, 2097970, 4647580, 33436508, 74070290, 532886746, 1180477648, 8492752016, 18813572666, 135351146098, 299836685596, 2157125586140, 4778573397458, 34378658232730, 76157337674320, 547901406138128, 1213738829392250

Offset: 1

Colin Barker, Oct 30 2015

			28 is in the sequence because (28+84)^3 - 28^3 = 1176^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,16,-16,-1,1).

Cf. A263942 (4), A263943 (21), A263944 (28), A263945 (39), A263946 (52), A263947 (57), A263948 (61) where the parenthesized number is k in the expression (n+k)^3 - n^3.

LinearRecurrence[{1, 16, -16, -1, 1}, {28, 476, 1106, 8218, 18256}, 30] (* Paolo Xausa, Mar 05 2024 *)

Vec(14*x*(x^4+4*x^3-13*x^2-32*x-2)/((x-1)*(x^4-16*x^2+1)) + O(x^40))

a(n) = a(n-1)+16*a(n-2)-16*a(n-3)-a(n-4)+a(n-5) for n>5.

G.f.: 14*x*(x^4+4*x^3-13*x^2-32*x-2) / ((x-1)*(x^4-16*x^2+1)).

cp's OEIS Frontend

A071954 a(n) = 4*a(n-1) - a(n-2) - 4, with a(0) = 2, a(1) = 4.

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A263943 Positive integers n such that (n+21)^3 - n^3 is a square.

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A263944 Positive integers n such that (n+28)^3 - n^3 is a square.

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A263945 Positive integers n such that (n+39)^3 - n^3 is a square.

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A263946 Positive integers n such that (n+52)^3 - n^3 is a square.

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A263947 Positive integers n such that (n+57)^3 - n^3 is a square.

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A263948 Positive integers n such that (n+61)^3 - n^3 is a square.

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