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A104563 A floretion-generated sequence relating to centered square numbers. 1
0, 1, 3, 5, 8, 13, 19, 25, 32, 41, 51, 61, 72, 85, 99, 113, 128, 145, 163, 181, 200, 221, 243, 265, 288, 313, 339, 365, 392, 421, 451, 481, 512, 545, 579, 613, 648, 685, 723, 761, 800, 841, 883, 925, 968, 1013, 1059, 1105, 1152, 1201, 1251 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

FORMULA

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

FAMP result: 2*a(n) + 2*A004525(n+1) = A104564(n) + a(n+1).

Superseeker results:

a(2*n+1) = A001844(n) = 2*n*(n+1) + 1 (Centered square numbers);

a(n+1) - a(n) = A098180(n) (Odd numbers with two times the odd numbers repeated in order between them);

a(n) + a(n+2) = A059100(n+1) = A010000(n+1);

a(n+2) - a(n) = A047599(n+1) (Numbers that are congruent to {0, 3, 4, 5} mod 8);

a(n+2) - 2*a(n+1) + a(n) = A007877(n+3) (Period 4 sequence with initial period (0, 1, 2, 1));

Coefficients of g.f.*(1-x)/(1+x) = convolution of this with A280560 gives A004525;

Coefficients of g.f./(1+x) = convolution of this with A033999 gives A054925.

a(n) = (1/2)*(n^2 + 1 - cos(n*Pi/2)). - Ralf Stephan, May 20 2007

From Colin Barker, Apr 29 2019: (Start)

a(n) = (2 - (-i)^n - i^n + 2*n^2) / 4 where i=sqrt(-1).

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

a(n) = A011848(n-1)+A011848(n+2). - R. J. Mathar, Sep 11 2019

PROG

Floretion Algebra Multiplication Program, FAMP Code: a(n) = 1vesrokseq[A*B] with A = - .5'i - .5i' + .5'ii' + .5e, B = + .5'ii' - .5'jj' + .5'kk' + .5e. RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code). Note: many slight variations of the "RokType" already exist, such that it has become difficult to assign them all names.

(PARI) concat(0, Vec(x*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ Colin Barker, Apr 29 2019

CROSSREFS

Cf. A001844, A004525, A104564, A098180, A059100, A010000, A047599, A007877.

Sequence in context: A053651 A175388 A310037 * A261175 A265064 A030762

Adjacent sequences:  A104560 A104561 A104562 * A104564 A104565 A104566

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Mar 15 2005

EXTENSIONS

Stephan's formula corrected by Bruno Berselli, Apr 29 2019

STATUS

approved

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Last modified April 7 01:03 EDT 2020. Contains 333291 sequences. (Running on oeis4.)